Let $f: X \subset \mathbb{R}^n \rightarrow \mathbb{R}$ be a strictly component-wise monotone function on the compact (but not convex) set $X \subset \mathbb{R}^n$.
I would like to interpolate $f$ on $X$, preserving the component-wise monotonicity, assuming I can only evaluate the function $f$ on a set of discrete points $C = \{x_i \in X\}$ that do not form a regular grid.
In one dimension, I know that the Monotone Cubic Interpolation, as well as linear interpolation would fulfill my requirements. As I do not have a regular grid, I cannot use repeated univariate interpolation.
Is there a multivariate interpolation method that preserves component-wise monotonicity and does not need a regular grid?
Would Polyharmonic Splines work for this (either in general, or for k=1)?